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Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

Bilender P. Allahverdiev, Hüseyin Tuna (2020)

Communications in Mathematics

In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

Summation equations with sign changing kernels and applications to discrete fractional boundary value problems

Christopher S. Goodrich (2016)

Commentationes Mathematicae Universitatis Carolinae

We consider the summation equation, for t [ μ - 2 , μ + b ] μ - 2 , y ( t ) = γ 1 ( t ) H 1 i = 1 n a i y ξ i + γ 2 ( t ) H 2 i = 1 m b i y ζ i + λ s = 0 b G ( t , s ) f ( s + μ - 1 , y ( s + μ - 1 ) ) in the case where the map ( t , s ) G ( t , s ) may change sign; here μ ( 1 , 2 ] is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that G is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions H 1 and H 2 . Finally, as an application of the abstract existence result,...

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